computer program to obtain ordinates for naca airfoilsuse of advanced nasa supercritical airfoil...
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National Aeronautics and Space AdministrationLangley Research Center • Hampton, Virginia 23681-0001
NASA Technical Memorandum 4741
Computer Program To Obtain Ordinates forNACA AirfoilsCharles L. Ladson, Cuyler W. Brooks, Jr., and Acquilla S. HillLangley Research Center • Hampton, Virginia
Darrell W. SprolesComputer Sciences Corporation • Hampton, Virginia
December 1996
Printed copies available from the following:
NASA Center for AeroSpace Information National Technical Information Service (NTIS)800 Elkridge Landing Road 5285 Port Royal RoadLinthicum Heights, MD 21090-2934 Springfield, VA 22161-2171(301) 621-0390 (703) 487-4650
The use of trademarks or names of manufacturers in this report is foraccurate reporting and does not constitute an official endorsement,either expressed or implied, of such products or manufacturers by theNational Aeronautics and Space Administration.
Available electronically at the following URL address: http://techreports.larc.nasa.gov/ltrs/ltrs.html
Computer program is available electronically from the Langley Software Server
Introduction
Although modern high-speed aircraft generally makeuse of advanced NASA supercritical airfoil sections,there is still a demand for information on the NACAseries of airfoil sections, which were developed over50years ago. Computer programs were developed in theearly 1970’s to produce the ordinates for airfoils of anythickness, thickness distribution, or camber in the NACAairfoil series. These programs are published in refer-ences 1 and 2. These programs, however, were written inthe Langley Research Center version of FORTRAN IVand are not easily portable to other computers. The pur-pose of this paper is to describe an updated version ofthese programs. The goal was to combine both programsinto a single program that could be executed on a widevariety of personal computers and workstations as wellas mainframes. The analytical design equations for bothsymmetrical and cambered airfoils in the NACA 4-digit-series, 4-digit-modified-series, 5-digit-series, 5-digit-modified-series, 16-series, 6-series, and 6A-series airfoilfamilies have been implemented. The camber-line desig-nations available are the 2-digit, 3-digit, 3-digit-reflex,6-series, and 6A-series. The program achieves portabilityby limiting machine-specific code. An effort was madeto make all inputs to the program as simple as possible touse and to lead the user through the process by means ofa menu.
Symbols
The symbols in parentheses are the ones used in thecomputer program and in the computer-generated listings(.rpt file).
A camber-line designation, fraction ofchord from leading edge over whichdesign load is uniform
ai constants in airfoil equation,i = 0, 1, ..., 4
bi constants in camber-line equation,i = 0,1, 2
c (C, CHD) airfoil chord
cl,i (CLI) design section lift coefficient
di constants in airfoil equation,i = 0, 1, 2, 3
dx derivative ofx; also basic selectableinterval in profile generation
d(x/c), dy, dδ derivatives ofx/c, y, andδI leading-edge radius index number
k1, k2 constants
m chordwise location for maximum ordi-nate of airfoil or camber line
p maximum ordinate of 2-digit camber line
R radius of curvature
Rle (RLE) leading-edge radius
r chordwise location for zero value of sec-ond derivative of 3-digit or 3-digit-reflexcamber-line equation
t thickness
x (X) distance along chord
y (Y) airfoil ordinate normal to chord, positiveabove chord
z complex variable in circle plane
z′ complex variable in near-circle plane
δ local inclination of camber line
ε airfoil parameter,φ − θζ complex variable in airfoil plane
θ angular coordinate ofz′φ angular coordinate ofz
ψ airfoil parameter determining radial coor-dinate ofz′
Abstract
Computer programs to produce the ordinates for airfoils of any thickness, thick-ness distribution, or camber in the NACA airfoil series were developed in the early1970’s and are published as NASA TM X-3069 and TM X-3284. For analytic airfoils,the ordinates are exact. For the 6-series and all but the leading edge of the 6A-seriesairfoils, agreement between the ordinates obtained from the program and previouslypublished ordinates is generally within 5× 10−5 chord. Since the publication of theseprograms, the use of personal computers and individual workstations has prolifer-ated. This report describes a computer program that combines the capabilities of thepreviously published versions. This program is written in ANSI FORTRAN 77 and canbe compiled to run on DOS, UNIX, and VMS based personal computers and work-stations as well as mainframes. An effort was made to make all inputs to the programas simple as possible to use and to lead the user through the process by means of amenu.
2
ψ0 average value of ψ,Subscripts:
cam cambered
l (L) lower surface
N forward portion of camber line
T aft portion of camber line
t thickness
u (U) upper surface
x derivative with respect tox
Computer Listing Symbols
For reasons having to do with code portability, thecomputer-generated listing (.rpt file) will always havethe alphabetic characters in upper case. The followinglist is intended to eliminate any confusion.
A camber-line designation, fraction of chordfrom leading edge over which design loadis uniform
A0,...,A4 constants in airfoil equation
CHD, C airfoil chord
CLI design section lift coefficient
CMB maximum camber in chord length,
CMBNMR number of camber lines to be combined in6- and 6A-series multiple camber-lineoption
CMY m, location of maximum camber
CRAT cumulative scaling of EPS, PSI,
D0,...,D3 constants in airfoil equation
DX basic selectable interval in profilegeneration
DY/DX first derivative ofy with respect tox,
D2Y/DX2 second derivative ofy with respect tox,
EPS airfoil parameter,ε = φ − θIT number of iterations to converge 6-series
profile
K1, K2 3-digit-reflex camber parameters,k1 andk2
PHI φ, angular coordinate ofz
PSI ψ, airfoil parameter determining radialcoordinate ofz′
RAT(I) ith iterative scaling ofε, ψRC radius of curvature at maximum thickness
for 4-digit modified profile
RK2OK1 3-digit reflex camber parameter ratio,
RLE leading-edge radius
RNP radius of curvature at origin
SF ratio of inputt/c to convergedt/c afterscaling
TOC thickness-chord ratio
X distance along chord
XMT m, location of maximum thickness for4-digit modified profile
XT(12), location and slope of ellipse nose fairingYT(12), for 6- and 6A-series thicknessYTP(12) profiles
XU, XL upper and lower surface locations ofx
XTP x/c location of slope sign change for6- and 6A-series thickness profile
XYM m, location of maximum thickness for6- and 6A-series profiles or chordwiselocation for maximum ordinate of airfoilor camber line
Y airfoil ordinate normal to chord, positiveabove chord
YM y/c location of slope sign change for 6- and6A-series thickness profile
YMAX y(m), maximum ordinate of thicknessdistribution
YU, YL upper and lower surfacey ordinate
Analysis
Thickness Distribution Equations for AnalyticAirfoils
The design equations for the analytic NACA airfoilsand camber lines have been presented in references 3to 7. They are repeated herein to provide a better under-standing of the computer program and indicate the use ofdifferent design variables. A summary of some of thedesign equations and ordinates for many airfoils fromthese families is also presented in references 8 to 10.
The traditional NACA airfoil designations are short-hand codes representing the essential elements (such asthickness-chord ratio, camber, design lift coefficient)controlling the shape of a profile generated within agiven airfoil type. Thus, for example the NACA 4-digit-series airfoil is specified by a 4-digit code of the formpmxx, wherep and m represent positions reserved for
12π------ ψ φd
0
2π
∫
py m( )
c------------=
Π RAT(I), I = 1 IT→
dydx------
d2y
dx2
--------
k2
k1-----
3
specification of the camber and xx allows for specifica-tion of the thickness-chord ratio as a percentage, that is,“pm12” designates a 12-percent-thick (t/c = 0.12) 4-digitairfoil.
NACA 4-digit-series airfoils.Symmetric airfoils inthe 4-digit-series family are designated by a 4-digit num-ber of the form NACA 00xx. The first two digits indicatea symmetric airfoil; the second two, the thickness-chordratio. Ordinates for the NACA 4-digit airfoil family(ref. 2) are described by an equation of the form:
The constants in the equation (fort/c = 0.20) weredetermined from the following boundary conditions:
Maximum ordinate:
Ordinate at trailing edge:
Magnitude of trailing-edge angle:
Nose shape:
The following coefficients were determined to meetthese constraints very closely:
To obtain ordinates for airfoils in the family with athickness other than 20 percent, the ordinates for themodel with a thickness-chord ratio of 0.20 are multipliedby the ratio (t/c)/0.20. The leading-edge radius ofthis family is defined as the radius of curvature of thebasic equation evaluated atx/c = 0. Because of the terma0(x/c)1/2 in the equation, the radius of curvature is finiteatx/c = 0 and can be shown to be (see appendix)
by taking the limit asx approaches zero of the standardexpression for radius of curvature:
To define an airfoil in this family, the only input neces-sary to the computer program is the desired thickness-chord ratio.
One might expect that this leading-edge radiusR(0),found in the limit as to depend only on thea0term of the defining equation, would also be the mini-mum radius on the profile curve. This is not true in gen-eral; for the NACA 0020 airfoil, for example, a slightlysmaller radius (R = 0.0435 as compared toR(0) =0.044075) is found in the vicinity ofx = 0.00025.
NACA 4-digit-modified-series airfoils.The 4-digit-modified-series airfoils are designated by a 4-digit num-ber followed by a dash and a 2-digit number (such asNACA 0012-63). The first two digits are zero for a sym-metrical airfoil and the second two digits indicate thethickness-chord ratio. The first digit after the dash is aleading-edge-radius index number, and the second is thelocation of maximum thickness in tenths of chord aft ofthe leading edge.
The design equation for the 4-digit-series airfoilfamily was modified (ref. 4) so that the same basic shapewas retained but variations in leading-edge radius andchordwise location of maximum thickness could bemade. Ordinates for these airfoils are determined fromthe following equations:
from leading edge to maximum thickness, and
from maximum thickness to trailing edge.
The constants in these equations (fort/c = 0.20) canbe determined from the following boundary conditions:
Maximum ordinate:
Leading-edge radius:
yc-- a0
xc--
1/2a1
xc--
a2xc--
2a3
xc--
3a4
xc--
4+ + + +=
xc-- 0.30= y
c-- 0.10= dy
dx------ 0=
xc-- 1.0= y
c-- 0.002=
xc-- 1.0= dy
dx------ 0.234=
xc-- 0.10= y
c-- 0.078=
a0 0.2969= a1 0.1260–=
a2 0.3516–= a3 0.2843=
a4 0.1015–=
Rle
a02
2----- t/c
0.20----------
2=
R1 dy/dx( )2
+[ ]3/2
d2y/dx
2-------------------------------------------=
x 0→
yc-- a0
xc--
1/2a1
xc--
a2xc--
2a3
xc--
3+ + +=
yc-- d0 d1 1 x
c--–
d2 1 xc--–
2d3 1 x
c--–
3+ + +=
xc-- m= y
c-- 0.10= dy
dx------ 0=
xc-- 0= R
a02
2-----=
4
Radius of curvature at maximum thickness:
Ordinate at trailing edge:
Magnitude of trailing-edge angle:
Thus, the maximum ordinate, slope, and radiusof curvature ofthe two portions of the surface match atx/c = m. The values ofd1 were chosen, as stated in refer-ence 4, to avoid reversals of curvature and are given inthe following table:
By use of these constraints, equations were writtenfor each of the constants (excepta0 andd1) in the equa-tion for the airfoil family and are included in the com-puter program. As in the 4-digit-series airfoil family,ordinates vary linearly with variations in thickness-chordratio and any desired thickness shape can be obtained byscaling the design ordinates by the ratio of the desiredthickness-chord ratio to the design thickness-chord ratio.
The leading-edge index is an arbitrary numberassigned to the leading-edge radius in reference 4 and isproportional toa0. The relationship between leading-edge radiusRle and index numberI is as follows:
Thus, an index of 0 indicates a sharp leading edge(radius of zero) and an index of 6 corresponds toa0 = 0.2969, the normal design value for the 20-percent-thick 4-digit airfoil. A value of leading-edge index of 9for a three times normal leading-edge radius was arbi-trarily assigned in reference 4, butI = 9 cannot be used inthis equation. In reference 4, the indexI is not usedin computation. Instead, the indexI = 9 was assigned toan airfoil where which(because ) thus has a three times normalleading-edge radius. The computer program is written sothat the desired value of leading-edge radius or the index
I is the input parameter. The value ofa0 is then computedin the program.
NACA 16-series airfoils.The NACA 16-series air-foil family is described in references 6 and 7. From theequation for the ordinates in reference 7, this series is aspecial case of the 4-digit-modified family although thisis not directly stated in the references. The 16-series air-foils are thus defined as having a leading-edge index of 4and a location of maximum thickness at 0.50 chord. Thedesignation NACA 16-012 airfoil is equivalent to anNACA 0012-45.
Thickness Distribution Equations for DerivedAirfoils
NACA 6-series airfoils.As described in references 9and 10, the basic symmetrical NACA 6-series airfoilswere developed by means of conformal transformations.The use of these transformations to relate the flow aboutan arbitrary airfoil to that of a near circle and then to acircle had been developed earlier, and the results are pre-sented in reference 11. The basic airfoil parametersψand ε are derived as a function ofφ, whereθ − φ isdefined as−ε. Figure 1,taken from reference 10, showsthe relationship between these variables in the complexplane. These parameters are used to compute both theairfoil ordinates and the potential flow velocity distribu-tion around the airfoil. For the NACA 6-series airfoils,the shape of the velocity distribution and the longitudinallocation of maximum velocity (or minimum pressure)were prescribed. The airfoil parametersψ and ε whichgive the desired velocity distribution were obtainedthrough an iterative process. Then the airfoil ordinatescan be calculated from these parameters by use of theequations presented in references 10 and 11. Thus, foreach prescribed velocity distribution, a set of basic airfoilparameters is obtained. However, as stated in refer-ence10, it is possible to define a set of basic parametersψ andε which can be multiplied by a constant factor toobtain airfoils of various thickness-chord ratios whilemaintaining the minimum pressure at the same chord-wise location. Thus, for each NACA 6-series airfoil fam-ily (i.e., 63-, 64-, or 65-series), there is one basic set ofvalues forψ andε.
NACA 6A-series airfoils.The NACA 6-series airfoilsections were developed in the early 1940’s. As aircraftspeeds increased, more attention was focused on the thin-ner airfoils of this series. However, difficulties wereencountered in the structural design and fabrication ofthese thinner sections because of the very thin trailingedges. As a result, the NACA 6A-series airfoil sectionswere developed, and details of these have been publishedin reference 12. Essentially, the modification consisted of
m d1
0.2 0.2000.3 0.2340.4 0.3150.5 0.4650.6 0.700
xc-- m= R
1 m–( )2
2d1 1 m–( ) 0.588–-----------------------------------------------=
xc-- 1.0= y
c-- d0 0.002= =
xc-- 1.0= dy
dx------ d1 f m( )= =
Rle 0.5 0.2969t/c0.2------- I
6---
2=
a0 0.2969 3 0.514246,= =Rle a0
2/2=
5
a near-constant slope from about the 80-percent-chordstation to the trailing edge and an increase in the trailing-edge thickness from zero to a finite value.
Calculation procedure.For each NACA 6-seriesairfoil family, a unique curve ofψ andε exists as a func-tion of φ. This curve can be scaled by a constant factor toprovide airfoils of different thickness within this family.A computer program could therefore be developed to cal-culate the airfoil ordinates for given values ofψ andε.Although the values of these basic airfoil parameterswere not published, tabulated values existed in files orcould be computed by the method of reference 11 frompublished airfoil ordinates. To provide more values ofψand ε for storage in a computer subroutine, a fit to theoriginal values was made with a parametric-linked cubic-spline-fit program, and nine additional values wereobtained between each of the original values. This pro-cess was carried out for each airfoil series, and the resultswere stored in the computer program as two arrays foreach airfoil family (one forε as a function ofφ and onefor ψ as a function ofφ). For the new program, the
enrichment was completely redone and theφ array isstored in the program only once for all 6- and 6A-seriesprofiles.
To calculate the ordinates for an arbitrary airfoil, theprogram first determines which airfoil series is desiredand calls for the subroutine for this series. The airfoilrepresented by the stored values ofψ andε is calculatedand its maximum thickness-chord ratio is determined.The ratio of the desired value to that obtained in thisdetermination is calculated. Then,ψ andε and are multi-plied by this ratio to arrive at a new airfoil thickness-chord ratio. The iteration is repeated until the computedthickness-chord ratio is within 0.01 percent of the desiredvalue or until 10 iterations have been performed. Usuallyconvergence occurs within four iterations. After the iter-ative process has converged within the limit established,any residual difference between the computed thickness-chord ratio and that desired is eliminated by linearly scal-ing they ordinate and its first and second derivatives bythe appropriate scale factor. The first and second deriva-tives of the airfoil ordinates as a function of chord arecomputed by a subroutine labeled “DERV” in the pro-gram. Although these ordinates and slopes are calculatedat more than 200 internally controlled chord stations, asubroutine is used to interpolate between these points (byuse of a piecewise cubic curve fit labeled “NTRP”) sothat the output will be in specified increments of chordstations.
Calculation of leading-edge radius.The values ofleading-edge radius of the derived airfoils, published inreferences 9 to 12, were initially determined by plottingthe ordinates to a large scale and fairing in the best circlefit by hand. Values of the tangency point between the cir-cle and airfoil surface obtained in this manner were notpublished. To provide smooth analytic ordinates aroundthe leading edge for the computer program, a tiltedellipse has been used. (See fig. 2.) This tilted ellipse isdescribed by the basic ellipse function plus an additiveterm, linear inx, which vanishes at the origin. The tiltedellipse expression thus has three arbitrary constants,which are determined in the procedure. The resulting fitto the airfoil ordinates is exact for the ordinate itself andthe first derivative and quite close for the second deriva-tive, though examination of the second derivative in theregion of tangency generally reveals a small discrepancy(even smaller in the current version than in the originalcomputer program (ref. 1)). The ellipse is defined so thatit has the same ordinate and slope as the airfoil surface atthe 12th (of 201) tabulated value ofφ in the airfoilparameter subroutine. (The 11th stored point, which isactually the second point of the original tabulated values,was used in ref. 1.) This tangency point is usually locatedat about 0.005 chord but varies with airfoilthickness and
Figure 1. Illustration of transformations used to derive airfoils andcalculate pressure distribution. From reference 1;a is basiclength, usually considered unity.
z-plane
= e(ψ–ψ0)+i (θ–φ)
z = aeψ0+i φ
z' = aeψ+i θ
z' z
z'-plane
ζ-plane
ζ = z' +
ζ = x + iy
a2
z'
aeψ0
aeψ
φ
θ
y
x
6
(a) 12-percent-thick example, NACA 64012 airfoil.
(b) 24-percent-thick example, NACA 64024 airfoil.
Figure 2. Leading-edge fairing for 6-series and 6A-series profiles.
6-series profile
Tilted ellipse
Curves match at 12th point
Ellipse, y(x)
Profile, y(x)
Radius, R(x)
.02
.01
0
.10
.05
0.010x/c
y(x) R(x)
.020
6-series profile
Tilted ellipse
Curves match at 12th point
Ellipse, y(x)
Profile, y(x)
Radius, R(x)
.03
.02
.01
0
.09
.06
.03
0.010 .010
x/c
y(x) R(x)
.020
7
series. By use of this method, a smooth transitionbetween airfoil and ellipse is produced, the tangencypoint is known, and there is a continuous variation ofleading-edge shape with thickness-chord ratio. The non-dimensional radius of curvature of the ellipse at the air-foil origin is also calculated in the program, and its valueis in close agreement with the published values of theleading-edge radius for known airfoils.
Camber-Line Equations
2-digit. The 2-digit camber line is designated by a2-digit number and, when used with a 4-digit airfoil,would have the form NACApmxx wherep is the maxi-mum camber in percent chord,m is the chordwise loca-tion of maximum camber in tenths of chord, and xx is theairfoil thickness in percent chord. The NACA 2-digitcamber line is described in reference 3. This camber lineis formed by two parabolic segments which have a gen-eral equation of the form:
The constants for the two equations are determinedfrom the following boundary conditions:
Camber-line extremities:
Maximum ordinate:
From these conditions, the camber-line equationsthen become
forward of the maximum ordinate and
aft of the maximum ordinate. Both the ordinate and slopeof the two parabolic segments match atx/c = m. Tablesof ordinates for some of these camber lines are tabulatedin references 9 and 10. The ordinates are linear withamount of camberp and these can be scaled up or downas desired.
3-digit. To provide a camber line with a very far for-ward location of the maximum camber, the 3-digit cam-ber line was developed and presented in reference 5. Thefirst digit of the 3-digit camber-line designation isdefined as two thirds of the design lift coefficient (intenths, i.e., 2 denotescl,i = 0.3); the second digit, as twicethe longitudinal location of maximum camber in tenthsof chord; and the third digit of zero indicates a non-reflexed trailing edge.
This camber line is also made up of two equations sothat the second derivative decreases to zero at a pointraft of the maximum ordinate and remains zero from thispoint to the trailing edge. The equations for these condi-tions are as follows:
from x/c = 0 tox/c = r, and
from x/c = r tox/c = 1.0. The boundary conditions are asfollows:
Camber-line extremities:
At the junction point:
The equation for the camber line then becomes
from x/c = 0 tox/c = r, and
from x/c = r to x/c = 1.0. These equations were thensolved for values ofr which would give longitudinallocations of the maximum ordinate of 5, 10, 15, 20, and25 percent chord. The value ofk1 was adjusted so that atheoretical design lift coefficient of 0.3 was obtained atthe ideal angle of attack. The value ofk1 can be linearlyscaled to give any desired design lift coefficient. Values
yc-- b0 b1
xc--
b2xc--
2+ +=
xc-- 0= y
c-- 0= x
c-- 1.0= y
c-- 0=
xc-- m= y
c-- p= dy
dx------ 0=
yc--
p
m2
------ 2mxc--
xc--
2–=
yc--
p
1 m–( )2-------------------- 1 2m–( ) 2m
xc-- x
c--
2–+=
d2y
dx2
--------- k1xc-- r–
=
d2y
dx2
--------- 0=
xc-- 0= y
c-- 0= x
c-- 1.0= y
c-- 0=
xc-- r=
yc--
N
yc--
T
=dydx------
N
dydx------
T
=
yc--
k1
6----- x
c--
33r
xc--
2– r
23 r–( )x
c--+=
yc--
k1r3
6---------- 1 x
c--–
=
8
of k1 and r and the camber-line designation were takenfrom reference 5 and are presented in the following table:
3-digit reflex.The camber-line designation for the3-digit-reflex camber line is the same as that for the3-digit camber line except that the last digit is changedfrom 0 to 1 to indicate the reflex characteristic, which isthe normally negative camber-line curvature, becomespositive in the aft segment.
For some applications, for example, rotorcraft mainrotors, to produce an airfoil with a quarter-chordpitching-moment coefficient of zero may be desirable.The 3-digit-reflexed camber line was thus designed tohave a theoretical zero pitching moment as described inreference 5. The forward part of the camber line is identi-cal to the 3-digit camber line but the aft portion waschanged from a zero curvature segment to a segmentwith positive curvature. The equation for the aft portionof the camber line is expressed by
By using the same boundary conditions as were used forthe 3-digit camber line, the equations for the ordinatesare
from x/c = 0 tox/c = r and
for x/c = r to x/c = 1.0. The ratiok2/k1 is expressed as
Values ofk1, k2/k1, andm for several camber-linedesignations from reference 5 are presented in the fol-lowing table:
6-series.The equations for the 6-series camber linesare presented in reference 9. The camber lines are a func-tion of the design lift coefficientcl,i and the chordwiseextent of uniform loadingA. The equation for these cam-ber lines is as follows:
where
As was true in reference 1, the program is capable ofcombining (by cumulative addition ofy/c) up to 10 cam-ber lines of this series to provide many types of loading.
16-series.The 16-series cambered airfoils, asdescribed in reference 6, are derived by using the 6-seriescamber-line equation with the mean-line loadingA setequal to 1.0, which is
This equation is the one for the standard mean linefor this series.
6A-series.The 6A-series cambered airfoils, asdescribed in reference 12, are derived by using a specialform of the 6-series camber-line equation. This specialform is designated as “theA = 0.8 modified mean line.”The modification basically consists of holding the slopeof the mean line constant from about the 85-percent-chord station to the trailing edge. As the reference indi-cates, this mean-line loading should always be used forthe 6A-series airfoils. This camber-line equation is given
Camber-linedesignation m r k1
210 0.05 0.0580 361.400220 0.10 0.1260 51.640230 0.15 0.2025 15.957240 0.20 0.2900 6.643250 0.25 0.3910 3.230
d2y
dx2
--------- k2xc-- r–
0>=xc-- r>
yc--
k1
6----- x
c-- r–
3 k2
k1----- 1 r–( )3 x
c--– r
3 xc--– r
3+=
yc--
k1
6-----
k2
k1----- x
c-- r–
3 k2
k1----- 1 r–( )3 x
c-- r
3 xc-- r
3+––=
k2
k1-----
31 r–----------- r m–( )2
r3
–=
Camber-linedesignation m r k1 k2/k1
221 0.10 0.1300 51.990 0.000764231 0.15 0.2170 15.793 0.00677241 0.20 0.3180 6.520 0.0303251 0.25 0.4410 3.191 0.1355
yc--
cl i,2π A 1+( )------------------------- 1
1 A–------------ 1
2--- A
xc--–
2log2 A
xc--–
=
12--- 1 x
c--–
2loge 1 x
c--–
–14--- 1 x
c--–
2+
14--- A
xc--–
2 xc-- loge
xc--– g h
xc--
–+–
g1
1 A–------------ A
2 12--- loge A
14---–
14---+–=
h1
1 A–------------ 1
2--- 1 A–( )2
loge 1 A–( ) 14--- 1 A–( )2
– g+=
yc--
cl i,4π-------- 1 x
c--–
2loge 1 x
c--–
=
9
as one of the options in selection of mean lines in theprogram.
Calculation of Cambered Airfoils
To calculate ordinates for a cambered airfoil (includ-ing the limiting cases of zero camber (ycam(x) = 0) andzero thickness (yt(x) = 0)), the desired mean (camber)line is first computed and then the ordinates of the sym-metrical airfoil are measured normal to the mean line atthe same chord station. This procedure leads to a set ofparametric equations where (y/c)t, (y/c)cam, andδ are allfunctions of the original independent variablex/c. Theupper surface ordinates on the cambered airfoil (x/c)uand (y/c)u are given by
where δ is the local inclination of the camber line and(y/c)t is assumed to be negative to obtain the lower sur-face ordinates (x/c)l and (y/c)l. This procedure is alsodescribed in reference 10.
The local slopes of the cambered airfoil are
and
by parametric differentiation of (x/c)u,l and (y/c)u,l withrespect to the originalx/c and use of the relationship
Although specific camber lines are generally usedwith specific thickness distributions, this program hasbeen written in a general format. As a result, any camberline can be used with either type thickness distribution sothat any shape desired can be generated.
The user should understand that these equations forcombining a thickness distribution with a mean line,while mathematically stable, can produce profiles that
are geometrically discontinuous and aerodynamicallyuseless. An NACA 2224 airfoil, for example, will have asmall region on the lower surface where (y/c)l is triplevalued.
The density of the final output of the procedure (as intables I and II) is determined by the user-supplied valueof the basicx/c interval dx. As the leading edge isapproached, the increments become smaller to providegood definition of the leading edge of a blunt profile.Ordinates are printed at increments ofdx/40 chord fromthe leading edge tox/c = 0.01250, at increments ofdx/4chord fromx/c = 0.01250 to 0.1000, and at increments ofdx chord fromx/c = 0.1000 to the trailing edge. The useris offered the option of settingdx and the smaller incre-ments forx/c less than 0.1 will be reset in the same ratio.This procedure is devised to provide rationalx values fordx = 0.01, 0.05, and 0.10.
Results and Discussion
Program Capabilities
AIRFOLS is the computer program which wasdeveloped to provide the airfoil shapes described in thesection “Analysis.” AIRFOLS, the NACA Airfoil Ordi-nate Generator, is the result of merging two previousefforts reported in references 1 and 2. This program cal-culates and then reports the ordinates and surface slopefor airfoils of any thickness, thickness distribution, orcamber in the designated NACA series. AIRFOLS is aportable, ANSI FORTRAN 77 code with limited plat-form dependencies. Parameters describing the desiredairfoil are entered into the software with menu andprompt driven input. Output consists of a report file and adata file.
Provisions have been made in the AIRFOLS pro-gram to combine basic airfoil shapes and camber linesfrom different series so that nonstandard as well as stan-dard airfoils can be generated. The analytical designequations for both symmetrical and cambered airfoils inthe NACA 4-digit-series, 4-digit-series modified, 5-digit-series, 5-digit-series modified, 16-series, 6-series, and6A-series airfoil families have been implemented. Thecamber-line designations available are the 2-digit,3-digit, 3-digit-reflex, 6-series, and 6A-series. The pro-gram achieves portability by limiting machine-specificcode. The two exceptions are a compiler-specific formatedit descriptor and a terminal-specific (display device)set of screen control codes. The user’s targets, selectedfrom the available platform configuration and terminaldevice choices, are communicated to the softwareenabling the machine-specific setup to be performedinternally with coded logic. The program attempts to becompletely self-guiding and to reduce required input to
xc--
u
xc--
yc--
tsin δ–=
yc--
u
yc--
cam
yc--
tcosδ+=
dydx------
u
δ secδtandydx------
t
yc--
t
dδd------ x
c--
tan δ–+
secδ dydx------
ttan δ–
yc--
t
dδd------ x
c--
–
---------------------------------------------------------------------------------------------=
dydx------
l
δ secδtandydx------
t
–yc--
t
dδd------ x
c--
tan δ+
secδ dydx------
ttan δ y
c--
t
dδd------ x
c--
+ +
---------------------------------------------------------------------------------------------=
dydx------
u
d y c⁄( )u/d x c⁄( )d x c⁄( )u/d x c⁄( )----------------------------------------=
10
the minimum necessary for the requested airfoil sectionand mean line. The convention used for the two outputfile names consist of combining a user-supplied file pre-fix respectively suffixed with “.rpt” and “.dta.” The out-put report (.rpt) is an ASCII file containing the tabulatednondimensional and dimensional airfoil ordinates as wellas other input and output parameters. The output data(.dta) is an ASCII file of X-Y pairs representing thedimensional airfoil ordinates.
To execute AIRFOLS, enter the appropriate com-mand then follow the program prompts. A messageprinted by the program at the beginning of executionshould reiterate the platform, operating system, andFORTRAN compiler to which the present executable istargeted. Examples of the supported platform configura-tion options are
CONVEX C2 using CONVEX FORTRAN compiler
IBM compatible PC using IBM FORTRAN/2compiler
Sun-3 using Sun FORTRAN compiler
VAX 780 using VAX FORTRAN compiler
IRIS-4D using Silicon Graphics FORTRANcompiler
Sun-4 using Sun FORTRAN compiler
Gateway 2000 PC using Lahey FORTRAN compiler
Generic, use only if other devices unavailable
The program’s initial request is for identification ofthe display device being used. Examples of the supporteddisplay device options are
ANSI terminal or emulator
IBM personal computer or compatible
TEKTRONIX 4107
Sun Microsystems workstation console
DEC VT terminal 50
DEC VT terminal 100
DEC VT terminal 200
DEC VT terminal 240 with REGIS
IBM personal computer VTERM package
There are two forms of input for controlling the AIR-FOLS program. The program prompts the user to entertextual information or select a choice from a menu. Theprimary technical input requests that a user sees are
Choice of standard or nonstandard airfoil
Selection from airfoil section options
Choice of airfoil with camber or no camber
Selection from mean-line options
Prompt for airfoil title
Prompt for output file prefix
Prompts for specific airfoil and mean-lineparameters
The program also makes available a nontechnicaltutorial in the form of a generic menu example to showthe use of menu control. To view the tutorial reply “F”(false) to the menu query: “USER IS EXPERIENCEDWITH AIRFOLS.” Replying to any menu query with“/EX” (respond with example) will also execute the tuto-rial. The tutorial interrupts the user’s interactive controlto execute the example menu type query. The desiredreplies are entered by the tutorial, alleviating require-ments for user interaction. Each reply produces a pro-gram response and a tutorial explanation of thatresponse. Program control is returned to the user whenthe example terminates. The tutorial example will dis-play the program’s response to a number of correct andincorrect entries. The responses to the program controlcharacters printed with each menu are covered as well.The program control characters and the typical responseto expect are
cr (or Enter) Clears screen then redisplays menu
/ Responds with system prompt optionsmenu
? Provides additional details on promptedrequest
E (or e) Exits from submenu to previous menulevel
Q (or q) Terminates program execution
AIRFOLS was written with emphasis on user friend-liness, self-direction, and informative error processingwith the goal of producing a valuable software tool forthe researcher.
Accuracy of Results
Analytical airfoils. All the airfoils and camber linesgenerated by this program are defined by closed analyti-cal expressions and no approximations have been madein the program. Thus, all results are exact. Many caseshave been run and compared with previously publishedresults to check the procedure, and for all cases thecomparisons were exact except for occasional differ-ences beyond the fifth digit caused by roundingdifferences.
11
Derived airfoils. In reference 1, about 25 cases,including several from each airfoil family, were com-puted for thickness-chord ratios from 0.06 to 0.15, andthe results were compared with the values published inreferences 9, 10, and 12. For the NACA 6-series airfoils,the agreement was generally within 5× 10-5 chord. TheNACA 6A-series airfoils show differences of as much as9 × 10−5 chord near the leading edge, but from aboutx/c= 0.10 to 0.95 the accuracy is about the same as forthe 6-series. The equations for the 6-series airfoil geome-try dictate that the trailing-edge thickness be zero; how-ever, the 6A-series airfoils have a finite trailing-edgethickness. For the 6A-series airfoils, the profiles obtainedfrom the program are not useful pastx/c = 0.95. For con-struction or grid generation, the user would probablymodify the last 5 percent of the chord by using the ordi-nate and slope atx/c = 0.95 and extrapolating to the trail-ing edge.
Early Release Version
A version of this program (0.A, March 31, 1992)already in use produces the same results for actual con-struction of an airfoil from the output coordinates. How-ever, this early version does display small discrepanciesin the second derivative of the thickness distributionupstream of 0.5 percent chord. In the current version, thisproblem has been corrected by allowing the tilted ellipsefit loop to go one step further and by replacing the inter-polation subroutine NTRP with an improved procedure.Users concerned about very small perturbations, as ingrid generation, may want to be sure they have the ver-sion documented in this report (1.01, April 1996).
Sample Output Tabulations
Sample computed ordinates for both a symmetricand a cambered airfoil are presented in tables I and II,
respectively. Printed at the top of the first page for eachtable is the airfoil and camber-line family selected, theairfoil designation, and a list of the input parameters forboth airfoil shape and camber line. Both nondimensionaland dimensional ordinates are listed. The dimensionalquantities have the same units as the input value of thechord, which is also listed at the top of the page. First andsecond derivatives of the surface ordinates are also pre-sented for symmetrical airfoils, but only first derivativesare tabulated for the cambered airfoils.
Concluding Remarks
Computer programs to produce the ordinates for air-foils of any thickness, thickness distribution, or camberin the NACA airfoil series were developed in the early1970’s and are published as NASA TM X-3069 andTM X-3284. For analytic airfoils, the ordinates are exact.For the 6-series and all but the leading edge of the 6A-series airfoils, agreement between the ordinates obtainedfrom the program and previously published ordinates isgenerally within 5× 10−5 chord. Since the publication ofthese programs, the use of personal computers andindividual workstations has proliferated. This reportdescribes a computer program that combines the capabil-ities of the previously published versions. This programis written in ANSI FORTRAN 77 and can be compiled torun on DOS, UNIX, and VMS based personal computersand workstations as well as mainframes. An effort wasmade to make all inputs to the program as simple as pos-sible to use and to lead the user through the process bymeans of a menu.
NASA Langley Research CenterHampton, VA 23681-0001August 29, 1996
12
Appendix
Convergence of Leading-Edge Radius ofNACA 4-Digit and 4-Digit-Modified SeriesAirfoils
The demonstration follows that in the limit as the radius of curvature at the nose of the NACA
4-digit-series airfoil profile (or the 4-digit-modified,which has the same equation at the leading edge) is finiteand equal to where is the coefficient of the
term in the defining equation:
The leading-edge radius is the radius of curvature at:
This expression for the radius of curvature cannot beevaluated atx = 0 by direct substitution. Evaluating thenumerator and denominator separately in the limit as
where all terms in must vanish and all con-stant terms become negligible compared with the termsin gives
For the numerator,
For the denominator,
Multiplying both numerator and denominator byx3
gives:
where the ellipses represent vanishing terms inThus,
This result clearly depends on thea0 term in the originalequation with its fractional power of the independentvariable. This term prevents the first and second deriva-tives from vanishing at the origin. Instead, both the firstand second derivatives increase without limit at the ori-gin but in such a way that the radius of curvature is finite.
As the variation ofR(x) is such that
so that even thoughR(0) is finite, Rx(0) is not. As aresult,R(x) does not appear to converge to until x isvery near zero, and then the curve forR(x) approaches
along the axisx = 0. Furthermore, becausea1 < 0(for 4-digit or 4-digit-modified airfoil),Rx(0) < 0 andthus R(x) must have a minimumRmin(x) < R(0). A gen-eral expression for this minimum is probably not worththe labor of the derivation. For the NACA 0020 airfoil, itis approximatelyRmin = R(0.000345) = 0.04304 com-pared with in nondimen-sional coordinates.
x 0,→
a02/2, a0
x1/2
y a0x1/2
a1x a2x2
a3x3
a4x4
+ + + +=
yx12---a0x
1/2–a1 2a2x 3a3x
24a4x
3+ + + +=
yxx14---a0x
3/2–2a2 6a3x 12a4x
2+ + +–=
x 0= Rle R 0( )=( )
R x( )1 yx
2+( )
3/2
yxx--------------------------=
R x( )2 1 yx2
+( )3
yxx2
----------------------
x=0
=
x 0,→ xe>0
xe<0,
1 yx2
+14---a0
2x
1–a1a0x
1/2–+=
1 yx2
+( )3 1
4---
3a0
6x
3–3
14---
2a1a0
5x
5/2–+=
314---
a12a0
4x
2–a1
3a0
3x
3/2–+ +
yxx2 1
4---
2a0
2x
3–2
12---
a2a0x3/2–
–=
232---
a3a0x1/2–
–
R x( )2 1/4( )3a0
63 1/4( )2
a1a05x
1/2 …+
1/4( )2a0
2 …----------------------------------------------------------------------------=
xe>0.
R 0( )2 14---a0
4=
R 0( ) 12---a0
2=
x 0,→
dRdx------- a1a0x
1/2–∼
a02/2
a02/2
R 0( ) a02/2 0.044075= =
13
References
1. Ladson, Charles L.; and Brooks, Cuyler W., Jr.: Developmentof a Computer Program To Obtain Ordinates for NACA 6- and6A-Series Airfoils. NASA TM X-3069, 1974.
2. Ladson, Charles L.; and Brooks, Cuyler W., Jr.: Developmentof a Computer Program To Obtain Ordinates for NACA4-Digit, 4-Digit Modified, 5-Digit, and 16-Series Airfoils.NASA TM-3284, 1975.
3. Jacobs, Eastman N.; Ward, Kenneth E.; and Pinkerton,Robert M.:The Characteristics of 78 Related Airfoil SectionsFrom Tests in the Variable-Density Wind Tunnel. NACARep. 460, 1933.
4. Stack, John; and Von Doenhoff, Albert E.:Tests of 16 RelatedAirfoils at High Speeds. NACA Rep. 492, 1934.
5. Jacobs, Eastman N.; and Pinkerton, Robert M.:Tests in theVariable-Density Wind Tunnel of Related Airfoils Having theMaximum Camber Unusually Far Forward. NACA Rep. 537,1935.
6. Stack, John:Tests of Airfoils Designed To Delay the Com-pressibility Burble. NACA Rep. 763, 1943.
7. Lindsey, W. F.; Stevenson, D. B.; and Daley, Bernard N.:Aero-dynamic Characteristics of 24 NACA 16-Series Airfoils atMach Numbers Between 0.3 and 0.8. NACA TN 1546, 1948.
8. Riegels, Friedrich Wilhelm (D. G. Randall, transl.):AerofoilSections. Butterworth & Co. (Publ.), Ltd., 1961.
9. Abbott, Ira H.; and Von Doenhoff, Albert E.: Theory of WingSections. Dover Publ., Inc., 1959.
10. Abbott, Ira H.; Von Doenhoff, Albert E.; and Stivers, Louis S.,Jr.:Summary of Airfoil Data. NACA Rep. 824, 1945.
11. Theodorsen, Theodore:Theory of Wing Sections of ArbitraryShape. NACA Rep. 411, 1931.
12. Loftin, Laurence K., Jr.: Theoretical and Experimental Datafor a Number of NACA 6A-Series Airfoil Sections. NACARep. 903, 1948.
14
Table I. Sample Computer File “64012.rpt” for Symmetric Airfoil
AIRFOLS ORDINATE INFORMATION REPORTfor 64012 airfoilPROFILE: 6-SERIES CAMBER: - none -PROFILE PARAMETERS:0.120000 = THICKNESS / CHORD (TOC)0.010000 = BASIC X INTERVAL (DX)6.000000 = MODEL CHORD (CHD)RAT(I) = 1.000000 0.581272 0.984284 0.9997004 = NUMBER OF ITERATIONS (IT)0.999994 = T/C RATIO, INPUT TO COMPUTED (SF)0.571965 = CUMULATIVE SCALING OF EPS,PSI (CRAT)0.060000 = MAXIMUM Y/C (YMAX)0.375975 = PEAK LOCATION (XYM)0.374842 0.060000 = SLOPE SIGN CHANGE LOCATION (XTP,YM)0.006717 = X/C FIT OF ELLIPSE (XT(12))0.011216 = Y/C FIT OF ELLIPSE (YT(12))0.780074 = SLOPE FIT OF ELLIPSE (YTP(12))0.010034 = RADIUS OF ELLIPSE, ORIGIN TO XT(12)/C,YT(12)/C (RNP)
X/C Y/C X Y DY/DX D2Y/DX20.000000 0.000000 0.000000 0.000000 ************* *************0.000250 0.002217 0.001500 0.013300 4.582653 -10133.4970700.000500 0.003153 0.003000 0.018918 3.151760 -3253.9919430.000750 0.003861 0.004500 0.023166 2.557617 -1787.5107420.001000 0.004446 0.006000 0.026674 2.218630 -1187.1180420.001250 0.004972 0.007500 0.029834 1.960144 -813.0159910.001500 0.005434 0.009000 0.032604 1.791810 -637.0929570.001750 0.005867 0.010500 0.035199 1.646746 -497.6483150.002000 0.006262 0.012000 0.037571 1.535877 -408.6473390.002250 0.006633 0.013500 0.039798 1.443982 -346.2497250.002500 0.006986 0.015000 0.041916 1.361467 -291.9172060.002750 0.007316 0.016500 0.043893 1.295341 -255.8052060.003000 0.007631 0.018000 0.045788 1.236006 -225.9541630.003250 0.007935 0.019500 0.047611 1.181388 -199.1908260.003500 0.008224 0.021000 0.049344 1.134461 -178.7955930.003750 0.008501 0.022500 0.051005 1.093128 -162.9010770.004000 0.008770 0.024000 0.052620 1.053807 -147.8429570.004250 0.009030 0.025500 0.054180 1.017726 -134.6956940.004500 0.009279 0.027000 0.055675 0.986034 -124.4665300.004750 0.009521 0.028500 0.057127 0.956632 -115.5008160.005000 0.009758 0.030000 0.058545 0.928518 -106.9959340.005250 0.009987 0.031500 0.059923 0.902220 -99.3903120.005500 0.010209 0.033000 0.061256 0.878265 -93.1223750.005750 0.010425 0.034500 0.062550 0.856161 -87.8770680.006000 0.010637 0.036000 0.063819 0.834813 -82.8085480.006250 0.010843 0.037500 0.065061 0.814474 -78.0150300.006500 0.011045 0.039000 0.066271 0.795407 -73.6028210.006750 0.011241 0.040500 0.067446 0.777865 -69.6733550.007000 0.011433 0.042000 0.068595 0.761235 -65.9774860.007250 0.011621 0.043500 0.069726 0.745170 -62.3765030.007500 0.011806 0.045000 0.070836 0.729842 -58.9413410.007750 0.011987 0.046500 0.071923 0.715419 -55.7429620.008000 0.012164 0.048000 0.072983 0.702071 -52.8523180.008250 0.012337 0.049500 0.074022 0.689594 -50.213284
15
Table I. Continued
0.008500 0.012508 0.051000 0.075048 0.677483 -47.6812250.008750 0.012676 0.052500 0.076058 0.665794 -45.3114970.009000 0.012842 0.054000 0.077051 0.654589 -43.1624340.009250 0.013004 0.055500 0.078026 0.643933 -41.2923660.009500 0.013163 0.057000 0.078981 0.633884 -39.7573170.009750 0.013320 0.058500 0.079921 0.624186 -38.3922650.010000 0.013475 0.060000 0.080850 0.614691 -37.0672840.012500 0.014909 0.075000 0.089454 0.536935 -25.5285280.015000 0.016179 0.090000 0.097075 0.482244 -18.9888610.017500 0.017331 0.105000 0.103985 0.441078 -14.0185460.020000 0.018394 0.120000 0.110366 0.410865 -10.4399530.022500 0.019392 0.135000 0.116349 0.387887 -8.0756120.025000 0.020338 0.150000 0.122027 0.369581 -6.7340700.027500 0.021242 0.165000 0.127449 0.353842 -5.8375920.030000 0.022109 0.180000 0.132653 0.340221 -5.1925670.032500 0.022944 0.195000 0.137662 0.327704 -4.7733600.035000 0.023748 0.210000 0.142490 0.316412 -4.2124040.037500 0.024527 0.225000 0.147161 0.306573 -3.7773470.040000 0.025282 0.240000 0.151690 0.297353 -3.6179450.042500 0.026014 0.255000 0.156082 0.288415 -3.5050870.045000 0.026724 0.270000 0.160345 0.279875 -3.3056940.047500 0.027414 0.285000 0.164481 0.271898 -3.0647290.050000 0.028084 0.300000 0.168504 0.264468 -2.9220850.052500 0.028736 0.315000 0.172417 0.257319 -2.8025850.055000 0.029371 0.330000 0.176224 0.250416 -2.6749420.057500 0.029989 0.345000 0.179933 0.243961 -2.5287630.060000 0.030591 0.360000 0.183545 0.237823 -2.3588850.062500 0.031178 0.375000 0.187069 0.232108 -2.1800800.065000 0.031752 0.390000 0.190512 0.226970 -2.0105490.067500 0.032313 0.405000 0.193879 0.222084 -1.9177790.070000 0.032862 0.420000 0.197174 0.217262 -1.9484370.072500 0.033400 0.435000 0.200398 0.212446 -1.8136310.075000 0.033925 0.450000 0.203550 0.208084 -1.6648150.077500 0.034440 0.465000 0.206640 0.204270 -1.5038160.080000 0.034946 0.480000 0.209678 0.200514 -1.4808950.082500 0.035443 0.495000 0.212660 0.196759 -1.5259020.085000 0.035930 0.510000 0.215580 0.193007 -1.4440460.087500 0.036408 0.525000 0.218446 0.189404 -1.3503030.090000 0.036878 0.540000 0.221268 0.186288 -1.2460980.092500 0.037340 0.555000 0.224037 0.183229 -1.1897650.095000 0.037794 0.570000 0.226761 0.180203 -1.1803590.097500 0.038241 0.585000 0.229444 0.177304 -1.1815160.100000 0.038681 0.600000 0.232084 0.174424 -1.1763150.110000 0.040368 0.660000 0.242209 0.163658 -0.9649890.120000 0.041955 0.720000 0.251727 0.153665 -0.9530140.130000 0.043446 0.780000 0.260673 0.144683 -0.8463580.140000 0.044851 0.840000 0.269105 0.136288 -0.8391680.150000 0.046174 0.900000 0.277045 0.128548 -0.7695060.160000 0.047421 0.960000 0.284524 0.120930 -0.7403890.170000 0.048594 1.020000 0.291562 0.113927 -0.6459120.180000 0.049701 1.080000 0.298208 0.107368 -0.6891260.190000 0.050742 1.140000 0.304450 0.100969 -0.5824940.200000 0.051721 1.200000 0.310329 0.094978 -0.631849
16
Table I. Continued
0.210000 0.052639 1.260000 0.315837 0.088733 -0.5915140.220000 0.053499 1.320000 0.320993 0.083079 -0.5604910.230000 0.054302 1.380000 0.325812 0.077596 -0.5697290.240000 0.055048 1.440000 0.330291 0.071720 -0.5677870.250000 0.055739 1.500000 0.334436 0.066492 -0.5256670.260000 0.056377 1.560000 0.338264 0.060994 -0.5698800.270000 0.056960 1.620000 0.341760 0.055913 -0.5085310.280000 0.057493 1.680000 0.344957 0.050420 -0.5570010.290000 0.057972 1.740000 0.347833 0.045442 -0.5069010.300000 0.058400 1.800000 0.350401 0.040442 -0.4742890.310000 0.058781 1.860000 0.352689 0.035488 -0.5001630.320000 0.059112 1.920000 0.354675 0.030786 -0.4650190.330000 0.059395 1.980000 0.356368 0.025781 -0.4881080.340000 0.059628 2.040000 0.357769 0.020854 -0.5341370.350000 0.059809 2.100000 0.358853 0.015181 -0.5835860.360000 0.059931 2.160000 0.359588 0.009271 -0.6186530.370000 0.059993 2.220000 0.359957 0.003000 -0.6173530.380000 0.059991 2.279999 0.359948 -0.003694 -0.7282680.390000 0.059916 2.339999 0.359499 -0.010928 -0.6860040.400000 0.059772 2.399999 0.358630 -0.018412 -0.7798220.410000 0.059548 2.459999 0.357290 -0.025435 -0.6577020.420000 0.059260 2.519999 0.355559 -0.032712 -0.7757430.430000 0.058897 2.579999 0.353381 -0.039309 -0.5989580.440000 0.058475 2.639999 0.350849 -0.045457 -0.6179730.450000 0.057991 2.699999 0.347946 -0.051196 -0.5672080.460000 0.057450 2.759999 0.344698 -0.056922 -0.5018780.470000 0.056858 2.819999 0.341145 -0.061447 -0.4667680.480000 0.056218 2.879999 0.337309 -0.066273 -0.4679650.490000 0.055534 2.939999 0.333204 -0.070572 -0.4449570.500000 0.054806 3.000000 0.328834 -0.075264 -0.4592140.510000 0.054032 3.060000 0.324192 -0.079229 -0.3602200.520000 0.053222 3.120000 0.319330 -0.082976 -0.4006050.530000 0.052372 3.180000 0.314233 -0.086970 -0.3698120.540000 0.051484 3.240000 0.308906 -0.090484 -0.3524210.550000 0.050561 3.300000 0.303368 -0.094048 -0.3107050.560000 0.049607 3.360000 0.297639 -0.096797 -0.2981870.570000 0.048621 3.420000 0.291726 -0.100354 -0.3642530.580000 0.047601 3.480000 0.285605 -0.103489 -0.2697710.590000 0.046553 3.539999 0.279317 -0.106118 -0.2996050.600000 0.045476 3.599999 0.272859 -0.109077 -0.2639430.610000 0.044374 3.659999 0.266245 -0.111324 -0.2291420.620000 0.043249 3.719999 0.259491 -0.113851 -0.2629630.630000 0.042098 3.779999 0.252586 -0.116378 -0.2376180.640000 0.040923 3.839999 0.245540 -0.118140 -0.1618410.650000 0.039733 3.899999 0.238400 -0.120230 -0.2344710.660000 0.038520 3.959999 0.231121 -0.122256 -0.1979080.670000 0.037287 4.019999 0.223724 -0.124178 -0.1423820.680000 0.036040 4.079999 0.216239 -0.125457 -0.1600290.690000 0.034776 4.139999 0.208654 -0.127102 -0.1429930.700000 0.033499 4.199999 0.200992 -0.128568 -0.1575160.710000 0.032206 4.259999 0.193236 -0.129713 -0.0765840.720000 0.030905 4.319999 0.185431 -0.130663 -0.1327820.730000 0.029592 4.379999 0.177554 -0.131876 -0.048956
17
Table I. Concluded
0.740000 0.028272 4.439999 0.169634 -0.132190 -0.0711990.750000 0.026946 4.499999 0.161675 -0.133017 -0.0144100.760000 0.025617 4.559999 0.153701 -0.132997 -0.0208500.770000 0.024285 4.619998 0.145708 -0.133368 0.0039550.780000 0.022952 4.679998 0.137712 -0.133111 0.0231930.790000 0.021622 4.739998 0.129730 -0.133023 0.0399570.800000 0.020294 4.799998 0.121766 -0.132388 0.0729030.810000 0.018974 4.859998 0.113846 -0.131488 0.1162630.820000 0.017665 4.919998 0.105989 -0.130493 0.1071310.830000 0.016366 4.979998 0.098198 -0.129051 0.1770690.840000 0.015084 5.039998 0.090504 -0.127493 0.1745490.850000 0.013819 5.099998 0.082913 -0.125490 0.2018940.860000 0.012574 5.159998 0.075443 -0.123375 0.2525690.870000 0.011354 5.219998 0.068122 -0.120714 0.2758180.880000 0.010160 5.279998 0.060962 -0.118055 0.3181840.890000 0.008996 5.339998 0.053979 -0.114893 0.3011350.900000 0.007864 5.399998 0.047186 -0.111270 0.3926890.910000 0.006772 5.459998 0.040631 -0.107152 0.4122450.920000 0.005721 5.519998 0.034324 -0.102876 0.5849720.930000 0.004720 5.579998 0.028319 -0.097767 0.4989440.940000 0.003771 5.639997 0.022626 -0.092050 0.5846600.950000 0.002884 5.699997 0.017303 -0.085061 0.7252630.960000 0.002073 5.759997 0.012437 -0.076628 0.9048500.970000 0.001351 5.819997 0.008106 -0.066925 1.0389570.980000 0.000739 5.879997 0.004435 -0.055016 1.3592240.990000 0.000264 5.939997 0.001581 -0.039331 1.9852441.000000 0.000000 5.999997 0.000000 -0.001854 21.079264
18
Tab
le II
. Sam
ple
Com
pute
r F
ile “
6441
2.rp
t” fo
r C
ambe
red
Airf
oil
AIR
FO
LS
OR
DIN
AT
E I
NF
OR
MA
TIO
N R
EP
OR
Tfo
r 6
44
12
airfo
ilP
RO
FIL
E:
6-S
ER
IES
CA
MB
ER
: 6
-SE
RIE
SP
RO
FIL
E P
AR
AM
ET
ER
S:
0.1
20
00
0
= T
HIC
KN
ES
S /
CH
OR
D
(TO
C)
0.0
10
00
0
= B
AS
IC X
IN
TE
RV
AL
(D
X)
6.0
00
00
0
= M
OD
EL
CH
OR
D
(CH
D)
RA
T(I
)
=
1
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00
00
0.5
81
27
2
0
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42
84
0.9
99
70
04
=
NU
MB
ER
OF
IT
ER
AT
ION
S
(IT
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99
94
=
T/C
RA
TIO
, IN
PU
T T
O C
OM
PU
TE
D
(SF
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19
65
=
CU
MU
LA
TIV
E S
CA
LIN
G O
F E
PS
,PS
I (
CR
AT
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00
00
=
MA
XIM
UM
Y/C
(Y
MA
X)
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75
97
5
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EA
K L
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AT
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CA
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M)
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06
71
7
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IT O
F E
LL
IPS
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(XT
(12
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12
16
=
Y/C
FIT
OF
EL
LIP
SE
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T(1
2))
0.7
80
07
4
= S
LO
PE
FIT
OF
EL
LIP
SE
(Y
TP
(12
))0
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00
34
=
RA
DIU
S O
F E
LL
IPS
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OR
IGIN
TO
XT
(12
)/C
,YT
(12
)/C
(R
NP
)C
AM
BE
R P
AR
AM
ET
ER
S:
1
= N
UM
BE
R O
F C
AM
BE
R L
INE
S T
O S
UM
(C
MB
NM
R)
CL
I(I)
=
0.4
00
00
0A
(I)
=
1
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00
00
UN
CA
MB
ER
ED
UP
PE
R S
UR
FA
CE
VA
LU
ES
LO
WE
R S
UR
FA
CE
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LU
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X
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X
U/C
Y
U/C
XU
YU
D
YU
/DX
U
X
L/C
Y
L/C
XL
YL
D
YL
/DX
L0
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00
00
0
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00
0
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00
00
0
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00
0
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00
00
-5
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94
16
0.0
00
00
0
0.0
00
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0
0.0
00
00
0
0.0
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00
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29
41
60
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02
50
-0
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01
19
0
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22
60
-0
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07
12
0
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35
58
1
9.7
01
13
8
0
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06
19
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.00
21
12
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37
12
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26
71
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88
01
0.0
00
50
0
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00
02
4
0.0
03
24
6
-0.0
00
14
6
0.0
19
47
6
6.8
84
83
1
0
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10
24
-0
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29
72
0
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61
46
-0
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78
33
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.92
85
16
0.0
00
75
0
0.0
00
10
8
0.0
04
00
3
0.0
00
64
8
0.0
24
01
7
4.6
81
64
0
0
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13
92
-0
.00
36
12
0
.00
83
52
-0
.02
16
70
-1
.65
11
75
0.0
01
00
0
0.0
00
26
1
0.0
04
63
5
0.0
01
56
5
0.0
27
81
3
3.7
33
71
6
0
.00
17
39
-0
.00
41
32
0
.01
04
35
-0
.02
47
92
-1
.47
50
05
0.0
01
25
0
0.0
00
42
3
0.0
05
20
9
0.0
02
54
0
0.0
31
25
3
3.1
17
45
5
0
.00
20
77
-0
.00
45
97
0
.01
24
60
-0
.02
75
84
-1
.33
06
05
0.0
01
50
0
0.0
00
59
7
0.0
05
71
7
0.0
03
57
9
0.0
34
30
0
2.7
56
69
4
0
.00
24
03
-0
.00
50
00
0
.01
44
21
-0
.03
00
02
-1
.23
14
27
0.0
01
75
0
0.0
00
77
5
0.0
06
19
4
0.0
04
64
8
0.0
37
16
5
2.4
68
12
0
0
.00
27
25
-0
.00
53
76
0
.01
63
52
-0
.03
22
54
-1
.14
24
63
0.0
02
00
0
0.0
00
95
9
0.0
06
63
4
0.0
05
75
4
0.0
39
80
4
2.2
60
17
2
0
.00
30
41
-0
.00
57
16
0
.01
82
46
-0
.03
42
93
-1
.07
21
66
0.0
02
25
0
0.0
01
14
7
0.0
07
04
9
0.0
06
88
4
0.0
42
29
3
2.0
95
46
0
0
.00
33
53
-0
.00
60
32
0
.02
01
16
-0
.03
61
95
-1
.01
23
17
0.0
02
50
0
0.0
01
33
9
0.0
07
44
5
0.0
08
03
2
0.0
44
67
1
1.9
53
10
6
0
.00
36
61
-0
.00
63
33
0
.02
19
68
-0
.03
79
95
-0
.95
73
06
0.0
02
75
0
0.0
01
53
4
0.0
07
81
7
0.0
09
20
4
0.0
46
90
4
1.8
42
61
6
0
.00
39
66
-0
.00
66
10
0
.02
37
96
-0
.03
96
61
-0
.91
23
26
0.0
03
00
0
0.0
01
73
2
0.0
08
17
5
0.0
10
39
0
0.0
49
05
2
1.7
46
07
8
0
.00
42
68
-0
.00
68
75
0
.02
56
10
-0
.04
12
51
-0
.87
12
68
0.0
03
25
0
0.0
01
93
1
0.0
08
52
1
0.0
11
58
7
0.0
51
12
5
1.6
59
30
9
0
.00
45
69
-0
.00
71
29
0
.02
74
13
-0
.04
27
73
-0
.83
28
75
0.0
03
50
0
0.0
02
13
3
0.0
08
85
1
0.0
12
79
9
0.0
53
10
5
1.5
86
30
8
0
.00
48
67
-0
.00
73
68
0
.02
92
01
-0
.04
42
11
-0
.79
94
20
0.0
03
75
0
0.0
02
33
7
0.0
09
16
9
0.0
14
02
4
0.0
55
01
1
1.5
23
16
3
0
.00
51
63
-0
.00
75
97
0
.03
09
76
-0
.04
55
80
-0
.76
95
89
0.0
04
00
0
0.0
02
54
3
0.0
09
47
8
0.0
15
25
6
0.0
56
86
9
1.4
64
06
7
0
.00
54
57
-0
.00
78
18
0
.03
27
44
-0
.04
69
08
-0
.74
08
86
0.0
04
25
0
0.0
02
74
9
0.0
09
77
8
0.0
16
49
7
0.0
58
66
9
1.4
10
65
9
0
.00
57
51
-0
.00
80
31
0
.03
45
03
-0
.04
81
84
-0
.71
42
66
0.0
04
50
0
0.0
02
95
8
0.0
10
06
7
0.0
17
74
9
0.0
60
40
3
1.3
64
37
8
0
.00
60
42
-0
.00
82
33
0
.03
62
51
-0
.04
94
00
-0
.69
06
59
0.0
04
75
0
0.0
03
16
8
0.0
10
34
9
0.0
19
00
8
0.0
62
09
1
1.3
21
96
0
0
.00
63
32
-0
.00
84
29
0
.03
79
92
-0
.05
05
75
-0
.66
85
66
0.0
05
00
0
0.0
03
37
9
0.0
10
62
4
0.0
20
27
3
0.0
63
74
3
1.2
08
26
1
0
.00
66
21
-0
.00
86
20
0
.03
97
27
-0
.05
17
19
-0
.69
18
87
0.0
05
25
0
0.0
03
60
6
0.0
10
89
5
0.0
21
63
4
0.0
65
36
9
1.1
72
25
2
0
.00
68
94
-0
.00
88
07
0
.04
13
66
-0
.05
28
42
-0
.67
21
41
19
Tab
le II
. Con
tinue
d
0.0
05
50
0
0.0
03
83
4
0.0
11
15
8
0.0
23
00
2
0.0
66
94
7
1.1
39
72
3
0
.00
71
66
-0
.00
89
87
0
.04
29
98
-0
.05
39
21
-0
.65
40
41
0.0
05
75
0
0.0
04
06
3
0.0
11
41
4
0.0
24
37
6
0.0
68
48
5
1.1
09
93
7
0
.00
74
37
-0
.00
91
61
0
.04
46
24
-0
.05
49
65
-0
.63
72
41
0.0
06
00
0
0.0
04
29
2
0.0
11
66
6
0.0
25
75
4
0.0
69
99
6
1.0
81
42
4
0
.00
77
08
-0
.00
93
31
0
.04
62
46
-0
.05
59
87
-0
.62
08
86
0.0
06
25
0
0.0
04
52
3
0.0
11
91
3
0.0
27
13
7
0.0
71
47
8
1.0
54
46
9
0
.00
79
77
-0
.00
94
97
0
.04
78
63
-0
.05
69
82
-0
.60
51
99
0.0
06
50
0
0.0
04
75
4
0.0
12
15
4
0.0
28
52
4
0.0
72
92
7
1.0
29
36
4
0
.00
82
46
-0
.00
96
58
0
.04
94
76
-0
.05
79
49
-0
.59
04
15
0.0
06
75
0
0.0
04
98
6
0.0
12
39
0
0.0
29
91
7
0.0
74
33
8
1.0
06
37
8
0
.00
85
14
-0
.00
98
14
0
.05
10
83
-0
.05
88
82
-0
.57
67
67
0.0
07
00
0
0.0
05
21
9
0.0
12
62
1
0.0
31
31
3
0.0
75
72
4
0.9
84
70
7
0
.00
87
81
-0
.00
99
65
0
.05
26
87
-0
.05
97
92
-0
.56
37
69
0.0
07
25
0
0.0
05
45
2
0.0
12
84
8
0.0
32
71
3
0.0
77
08
8
0.9
63
90
1
0
.00
90
48
-0
.01
01
14
0
.05
42
87
-0
.06
06
85
-0
.55
11
45
0.0
07
50
0
0.0
05
68
6
0.0
13
07
2
0.0
34
11
5
0.0
78
43
0
0.9
44
14
9
0
.00
93
14
-0
.01
02
60
0
.05
58
85
-0
.06
15
59
-0
.53
90
48
0.0
07
75
0
0.0
05
92
0
0.0
13
29
1
0.0
35
52
2
0.0
79
74
8
0.9
25
63
6
0
.00
95
80
-0
.01
04
02
0
.05
74
78
-0
.06
24
12
-0
.52
76
31
0.0
08
00
0
0.0
06
15
5
0.0
13
50
6
0.0
36
93
1
0.0
81
03
8
0.9
08
54
5
0
.00
98
45
-0
.01
05
40
0
.05
90
69
-0
.06
32
40
-0
.51
70
52
0.0
08
25
0
0.0
06
39
1
0.0
13
71
8
0.0
38
34
4
0.0
82
30
5
0.8
92
60
8
0
.01
01
09
-0
.01
06
75
0
.06
06
56
-0
.06
40
49
-0
.50
71
51
0.0
08
50
0
0.0
06
62
7
0.0
13
92
6
0.0
39
75
9
0.0
83
55
7
0.8
77
20
5
0
.01
03
73
-0
.01
08
07
0
.06
22
41
-0
.06
48
45
-0
.49
75
03
0.0
08
75
0
0.0
06
86
3
0.0
14
13
2
0.0
41
17
7
0.0
84
79
3
0.8
62
39
5
0
.01
06
37
-0
.01
09
38
0
.06
38
23
-0
.06
56
27
-0
.48
81
60
0.0
09
00
0
0.0
07
09
9
0.0
14
33
5
0.0
42
59
6
0.0
86
01
0
0.8
48
24
2
0
.01
09
01
-0
.01
10
66
0
.06
54
04
-0
.06
63
94
-0
.47
91
81
0.0
09
25
0
0.0
07
33
6
0.0
14
53
5
0.0
44
01
8
0.0
87
20
8
0.8
34
81
4
0
.01
11
64
-0
.01
11
91
0
.06
69
82
-0
.06
71
44
-0
.47
06
28
0.0
09
50
0
0.0
07
57
4
0.0
14
73
1
0.0
45
44
3
0.0
88
38
5
0.8
22
17
1
0
.01
14
26
-0
.01
13
13
0
.06
85
57
-0
.06
78
76
-0
.46
25
57
0.0
09
75
0
0.0
07
81
2
0.0
14
92
4
0.0
46
87
0
0.0
89
54
6
0.8
10
00
3
0
.01
16
88
-0
.01
14
32
0
.07
01
30
-0
.06
85
95
-0
.45
47
51
0.0
10
00
0
0.0
08
05
0
0.0
15
11
6
0.0
48
29
9
0.0
90
69
5
0.7
98
13
0
0
.01
19
50
-0
.01
15
51
0
.07
17
01
-0
.06
93
04
-0
.44
70
82
0.0
12
50
0
0.0
10
44
6
0.0
16
90
6
0.0
62
67
7
0.1
01
43
5
0.7
01
78
7
0
.01
45
54
-0
.01
26
28
0
.08
73
23
-0
.07
57
68
-0
.38
38
74
0.0
15
00
0
0.0
12
86
4
0.0
18
51
7
0.0
77
18
3
0.1
11
10
0
0.6
34
60
6
0
.01
71
36
-0
.01
35
58
0
.10
28
17
-0
.08
13
51
-0
.33
92
60
0.0
17
50
0
0.0
15
29
6
0.0
19
99
6
0.0
91
77
6
0.1
19
97
5
0.5
84
20
1
0
.01
97
04
-0
.01
43
84
0
.11
82
24
-0
.08
63
06
-0
.30
57
22
0.0
20
00
0
0.0
17
73
9
0.0
21
37
5
0.1
06
43
1
0.1
28
25
3
0.5
46
90
3
0
.02
22
61
-0
.01
51
34
0
.13
35
68
-0
.09
08
05
-0
.28
14
86
0.0
22
50
0
0.0
20
18
9
0.0
22
67
9
0.1
21
13
2
0.1
36
07
3
0.5
18
19
4
0
.02
48
11
-0
.01
58
28
0
.14
88
68
-0
.09
49
67
-0
.26
34
26
0.0
25
00
0
0.0
22
64
4
0.0
23
92
2
0.1
35
86
6
0.1
43
53
3
0.4
95
05
1
0
.02
73
56
-0
.01
64
80
0
.16
41
34
-0
.09
88
78
-0
.24
93
29
0.0
27
50
0
0.0
25
10
4
0.0
25
11
5
0.1
50
62
7
0.1
50
68
9
0.4
75
07
4
0
.02
98
96
-0
.01
70
97
0
.17
93
73
-0
.10
25
83
-0
.23
73
16
0.0
30
00
0
0.0
27
56
9
0.0
26
26
4
0.1
65
41
1
0.1
57
58
2
0.4
57
68
9
0
.03
24
31
-0
.01
76
86
0
.19
45
89
-0
.10
61
14
-0
.22
70
34
0.0
32
50
0
0.0
30
03
6
0.0
27
37
3
0.1
80
21
6
0.1
64
23
9
0.4
41
75
5
0
.03
49
64
-0
.01
82
49
0
.20
97
84
-0
.10
94
92
-0
.21
75
65
0.0
35
00
0
0.0
32
50
7
0.0
28
44
6
0.1
95
04
0
0.1
70
67
8
0.4
27
35
3
0
.03
74
93
-0
.01
87
88
0
.22
49
60
-0
.11
27
27
-0
.20
90
67
0.0
37
50
0
0.0
34
98
0
0.0
29
48
7
0.2
09
87
9
0.1
76
92
4
0.4
14
68
2
0
.04
00
20
-0
.01
93
07
0
.24
01
21
-0
.11
58
40
-0
.20
17
91
0.0
40
00
0
0.0
37
45
5
0.0
30
49
9
0.2
24
73
3
0.1
82
99
4
0.4
02
83
3
0
.04
25
45
-0
.01
98
07
0
.25
52
67
-0
.11
88
44
-0
.19
49
59
0.0
42
50
0
0.0
39
93
3
0.0
31
48
3
0.2
39
60
0
0.1
88
89
8
0.3
91
43
4
0
.04
50
67
-0
.02
02
91
0
.27
04
00
-0
.12
17
43
-0
.18
82
66
0.0
45
00
0
0.0
42
41
3
0.0
32
44
0
0.2
54
48
0
0.1
94
64
2
0.3
80
58
6
0
.04
75
87
-0
.02
07
57
0
.28
55
20
-0
.12
45
42
-0
.18
18
36
0.0
47
50
0
0.0
44
89
5
0.0
33
37
2
0.2
69
37
3
0.2
00
23
2
0.3
70
44
6
0
.05
01
05
-0
.02
12
07
0
.30
06
27
-0
.12
72
42
-0
.17
58
46
0.0
50
00
0
0.0
47
37
9
0.0
34
28
0
0.2
84
27
6
0.2
05
68
3
0.3
60
98
3
0
.05
26
21
-0
.02
16
43
0
.31
57
24
-0
.12
98
55
-0
.17
02
91
0.0
52
50
0
0.0
49
86
5
0.0
35
16
6
0.2
99
18
9
0.2
10
99
8
0.3
51
90
9
0
.05
51
35
-0
.02
20
64
0
.33
08
11
-0
.13
23
84
-0
.16
49
22
0.0
55
00
0
0.0
52
35
2
0.0
36
03
1
0.3
14
11
3
0.2
16
18
3
0.3
43
17
7
0
.05
76
48
-0
.02
24
72
0
.34
58
87
-0
.13
48
30
-0
.15
97
14
0.0
57
50
0
0.0
54
84
1
0.0
36
87
5
0.3
29
04
5
0.2
21
24
7
0.3
34
98
8
0
.06
01
59
-0
.02
28
67
0
.36
09
55
-0
.13
72
01
-0
.15
48
72
0.0
60
00
0
0.0
57
33
1
0.0
37
69
9
0.3
43
98
6
0.2
26
19
3
0.3
27
20
0
0
.06
26
69
-0
.02
32
50
0
.37
60
14
-0
.13
94
98
-0
.15
02
74
0.0
62
50
0
0.0
59
82
2
0.0
38
50
5
0.3
58
93
4
0.2
31
02
8
0.3
19
91
3
0
.06
51
78
-0
.02
36
21
0
.39
10
66
-0
.14
17
27
-0
.14
60
30
0.0
65
00
0
0.0
62
31
5
0.0
39
29
4
0.3
73
89
0
0.2
35
76
3
0.3
13
27
9
0
.06
76
85
-0
.02
39
83
0
.40
61
10
-0
.14
38
95
-0
.14
22
97
0.0
67
50
0
0.0
64
80
9
0.0
40
06
7
0.3
88
85
2
0.2
40
40
3
0.3
06
95
8
0
.07
01
91
-0
.02
43
35
0
.42
11
48
-0
.14
60
08
-0
.13
87
61
0.0
70
00
0
0.0
67
30
3
0.0
40
82
5
0.4
03
82
0
0.2
44
95
1
0.3
00
75
4
0
.07
26
97
-0
.02
46
78
0
.43
61
80
-0
.14
80
68
-0
.13
52
43
0.0
72
50
0
0.0
69
79
9
0.0
41
56
8
0.4
18
79
4
0.2
49
40
9
0.2
94
60
4
0
.07
52
01
-0
.02
50
12
0
.45
12
06
-0
.15
00
74
-0
.13
16
86
0.0
75
00
0
0.0
72
29
6
0.0
42
29
6
0.4
33
77
4
0.2
53
77
8
0.2
88
96
2
0
.07
77
04
-0
.02
53
38
0
.46
62
26
-0
.15
20
26
-0
.12
85
35
0.0
77
50
0
0.0
74
79
3
0.0
43
01
1
0.4
48
75
9
0.2
58
06
8
0.2
83
92
0
0
.08
02
07
-0
.02
56
56
0
.48
12
41
-0
.15
39
35
-0
.12
58
88
20
Tab
le II
. Con
tinue
d
0.0
80
00
0
0.0
77
29
1
0.0
43
71
5
0.4
63
74
8
0.2
62
28
8
0.2
78
97
5
0
.08
27
09
-0
.02
59
68
0
.49
62
52
-0
.15
58
06
-0
.12
32
63
0.0
82
50
0
0.0
79
79
0
0.0
44
40
6
0.4
78
74
2
0.2
66
43
7
0.2
74
06
6
0
.08
52
10
-0
.02
62
73
0
.51
12
58
-0
.15
76
38
-0
.12
06
08
0.0
85
00
0
0.0
82
29
0
0.0
45
08
5
0.4
93
74
0
0.2
70
50
7
0.2
69
19
3
0
.08
77
10
-0
.02
65
71
0
.52
62
60
-0
.15
94
24
-0
.11
79
23
0.0
87
50
0
0.0
84
79
0
0.0
45
75
1
0.5
08
74
3
0.2
74
50
9
0.2
64
50
3
0
.09
02
10
-0
.02
68
62
0
.54
12
57
-0
.16
11
72
-0
.11
53
58
0.0
90
00
0
0.0
87
29
1
0.0
46
40
8
0.5
23
74
9
0.2
78
45
0
0.2
60
33
8
0
.09
27
09
-0
.02
71
48
0
.55
62
51
-0
.16
28
90
-0
.11
32
47
0.0
92
50
0
0.0
89
79
3
0.0
47
05
4
0.5
38
75
8
0.2
82
32
6
0.2
56
25
8
0
.09
52
07
-0
.02
74
28
0
.57
12
41
-0
.16
45
70
-0
.11
11
67
0.0
95
00
0
0.0
92
29
5
0.0
47
69
0
0.5
53
77
2
0.2
86
14
1
0.2
52
23
7
0
.09
77
05
-0
.02
77
03
0
.58
62
28
-0
.16
62
19
-0
.10
90
96
0.0
97
50
0
0.0
94
79
8
0.0
48
31
7
0.5
68
78
8
0.2
89
90
1
0.2
48
36
8
0
.10
02
02
-0
.02
79
73
0
.60
12
12
-0
.16
78
40
-0
.10
71
28
0.1
00
00
0
0.0
97
30
1
0.0
48
93
4
0.5
83
80
8
0.2
93
60
4
0.2
44
54
4
0
.10
26
99
-0
.02
82
39
0
.61
61
92
-0
.16
94
32
-0
.10
51
58
0.1
10
00
0
0.1
07
31
9
0.0
51
30
9
0.6
43
91
6
0.3
07
85
4
0.2
30
21
6
0
.11
26
81
-0
.02
92
49
0
.67
60
83
-0
.17
54
95
-0
.09
78
27
0.1
20
00
0
0.1
17
34
5
0.0
53
55
0
0.7
04
06
7
0.3
21
30
0
0.2
16
95
9
0
.12
26
55
-0
.03
01
91
0
.73
59
33
-0
.18
11
45
-0
.09
09
92
0.1
30
00
0
0.1
27
37
6
0.0
55
66
5
0.7
64
25
5
0.3
33
99
2
0.2
04
96
3
0
.13
26
24
-0
.03
10
67
0
.79
57
44
-0
.18
64
03
-0
.08
49
36
0.1
40
00
0
0.1
37
41
3
0.0
57
66
6
0.8
24
47
6
0.3
45
99
9
0.1
93
76
3
0
.14
25
87
-0
.03
18
86
0
.85
55
23
-0
.19
13
14
-0
.07
92
73
0.1
50
00
0
0.1
47
45
4
0.0
59
55
9
0.8
84
72
6
0.3
57
35
6
0.1
83
39
5
0
.15
25
46
-0
.03
26
49
0
.91
52
73
-0
.19
58
93
-0
.07
40
97
0.1
60
00
0
0.1
57
50
0
0.0
61
35
0
0.9
45
00
3
0.3
68
10
0
0.1
73
30
2
0
.16
25
00
-0
.03
33
60
0
.97
49
97
-0
.20
01
58
-0
.06
89
01
0.1
70
00
0
0.1
67
55
0
0.0
63
04
3
1.0
05
30
3
0.3
78
26
0
0.1
63
95
6
0
.17
24
49
-0
.03
40
21
1
.03
46
97
-0
.20
41
24
-0
.06
41
93
0.1
80
00
0
0.1
77
60
4
0.0
64
64
8
1.0
65
62
3
0.3
87
89
1
0.1
55
17
1
0
.18
23
96
-0
.03
46
39
1
.09
43
77
-0
.20
78
32
-0
.05
98
21
0.1
90
00
0
0.1
87
66
1
0.0
66
16
5
1.1
25
96
3
0.3
96
98
8
0.1
46
64
8
0
.19
23
39
-0
.03
52
11
1
.15
40
37
-0
.21
12
64
-0
.05
55
11
0.2
00
00
0
0.1
97
72
0
0.0
67
59
9
1.1
86
31
9
0.4
05
59
7
0.1
38
62
4
0
.20
22
80
-0
.03
57
43
1
.21
36
81
-0
.21
44
57
-0
.05
15
25
0.2
10
00
0
0.2
07
78
2
0.0
68
95
2
1.2
46
69
2
0.4
13
71
5
0.1
30
42
6
0
.21
22
18
-0
.03
62
33
1
.27
33
08
-0
.21
73
98
-0
.04
72
04
0.2
20
00
0
0.2
17
84
6
0.0
70
22
7
1.3
07
07
9
0.4
21
36
5
0.1
22
89
3
0
.22
21
54
-0
.03
66
83
1
.33
29
22
-0
.22
01
01
-0
.04
34
08
0.2
30
00
0
0.2
27
91
3
0.0
71
42
8
1.3
67
47
8
0.4
28
56
6
0.1
15
59
4
0
.23
20
87
-0
.03
70
96
1
.39
25
22
-0
.22
25
77
-0
.03
97
20
0.2
40
00
0
0.2
37
98
2
0.0
72
55
3
1.4
27
89
0
0.4
35
31
7
0.1
07
96
4
0
.24
20
18
-0
.03
74
70
1
.45
21
11
-0
.22
48
20
-0
.03
55
81
0.2
50
00
0
0.2
48
05
2
0.0
73
60
5
1.4
88
31
2
0.4
41
63
0
0.1
01
03
2
0
.25
19
48
-0
.03
78
06
1
.51
16
88
-0
.22
68
34
-0
.03
20
40
0.2
60
00
0
0.2
58
12
4
0.0
74
58
7
1.5
48
74
4
0.4
47
52
2
0.0
93
88
1
0
.26
18
76
-0
.03
81
05
1
.57
12
56
-0
.22
86
30
-0
.02
81
81
0.2
70
00
0
0.2
68
19
8
0.0
75
49
7
1.6
09
18
5
0.4
52
98
3
0.0
87
18
9
0
.27
18
02
-0
.03
83
66
1
.63
08
15
-0
.23
01
95
-0
.02
46
99
0.2
80
00
0
0.2
78
27
2
0.0
76
34
1
1.6
69
63
4
0.4
58
04
7
0.0
80
12
8
0
.28
17
28
-0
.03
85
93
1
.69
03
66
-0
.23
15
55
-0
.02
07
63
0.2
90
00
0
0.2
88
34
8
0.0
77
11
6
1.7
30
09
0
0.4
62
69
4
0.0
73
61
6
0
.29
16
52
-0
.03
87
82
1
.74
99
09
-0
.23
26
89
-0
.01
73
10
0.3
00
00
0
0.2
98
42
5
0.0
77
82
3
1.7
90
55
3
0.4
66
94
0
0.0
67
11
6
0
.30
15
74
-0
.03
89
34
1
.80
94
47
-0
.23
36
07
-0
.01
38
02
0.3
10
00
0
0.3
08
50
3
0.0
78
46
9
1.8
51
02
0
0.4
70
81
4
0.0
60
69
2
0
.31
14
97
-0
.03
90
56
1
.86
89
79
-0
.23
43
35
-0
.01
03
11
0.3
20
00
0
0.3
18
58
2
0.0
79
04
9
1.9
11
49
2
0.4
74
29
6
0.0
54
54
7
0
.32
14
18
-0
.03
91
42
1
.92
85
07
-0
.23
48
49
-0
.00
70
47
0.3
30
00
0
0.3
28
66
1
0.0
79
56
6
1.9
71
96
8
0.4
77
39
7
0.0
48
12
6
0
.33
13
38
-0
.03
91
93
1
.98
80
31
-0
.23
51
58
-0
.00
34
53
0.3
40
00
0
0.3
38
74
1
0.0
80
02
0
2.0
32
44
8
0.4
80
11
8
0.0
41
80
7
0
.34
12
59
-0
.03
92
10
2
.04
75
52
-0
.23
52
61
0
.00
00
86
0.3
50
00
0
0.3
48
82
2
0.0
80
40
6
2.0
92
93
0
0.4
82
43
7
0.0
34
76
7
0
.35
11
78
-0
.03
91
88
2
.10
70
69
-0
.23
51
30
0
.00
43
97
0.3
60
00
0
0.3
58
90
3
0.0
80
72
0
2.1
53
41
5
0.4
84
32
2
0.0
27
51
3
0
.36
10
97
-0
.03
91
22
2
.16
65
84
-0
.23
47
34
0
.00
89
66
0.3
70
00
0
0.3
68
98
4
0.0
80
95
9
2.2
13
90
2
0.4
85
75
6
0.0
19
91
7
0
.37
10
16
-0
.03
90
09
2
.22
60
97
-0
.23
40
54
0
.01
39
16
0.3
80
00
0
0.3
79
06
5
0.0
81
12
2
2.2
74
39
1
0.4
86
73
1
0.0
11
91
9
0
.38
09
35
-0
.03
88
46
2
.28
56
08
-0
.23
30
78
0
.01
93
07
0.3
90
00
0
0.3
89
14
7
0.0
81
19
7
2.3
34
88
1
0.4
87
18
4
0.0
03
39
8
0
.39
08
53
-0
.03
86
23
2
.34
51
18
-0
.23
17
41
0
.02
52
57
0.4
00
00
0
0.3
99
22
9
0.0
81
18
9
2.3
95
37
1
0.4
87
13
6
-0.0
05
35
8
0
.40
07
71
-0
.03
83
44
2
.40
46
28
-0
.23
00
65
0
.03
14
71
0.4
10
00
0
0.4
09
31
0
0.0
81
08
9
2.4
55
86
0
0.4
86
53
7
-0.0
13
64
7
0
.41
06
90
-0
.03
79
99
2
.46
41
38
-0
.22
79
96
0
.03
72
31
0.4
20
00
0
0.4
19
39
1
0.0
80
91
1
2.5
16
34
6
0.4
85
46
6
-0.0
22
17
8
0
.42
06
09
-0
.03
76
02
2
.52
36
52
-0
.22
56
14
0
.04
32
55
0.4
30
00
0
0.4
29
47
1
0.0
80
64
5
2.5
76
82
9
0.4
83
87
0
-0.0
30
02
7
0
.43
05
28
-0
.03
71
44
2
.58
31
69
-0
.22
28
63
0
.04
85
99
0.4
40
00
0
0.4
39
55
1
0.0
80
30
7
2.6
37
30
6
0.4
81
84
1
-0.0
37
42
6
0
.44
04
49
-0
.03
66
39
2
.64
26
92
-0
.21
98
35
0
.05
34
97
0.4
50
00
0
0.4
49
62
9
0.0
79
89
4
2.6
97
77
7
0.4
79
36
3
-0.0
44
41
4
0
.45
03
70
-0
.03
60
86
2
.70
22
21
-0
.21
65
14
0
.05
79
86
0.4
60
00
0
0.4
59
70
7
0.0
79
41
1
2.7
58
24
0
0.4
76
46
3
-0.0
51
38
6
0
.46
02
93
-0
.03
54
87
2
.76
17
58
-0
.21
29
24
0
.06
24
65
0.4
70
00
0
0.4
69
78
2
0.0
78
86
3
2.8
18
69
4
0.4
73
18
0
-0.0
57
16
5
0
.47
02
17
-0
.03
48
51
2
.82
13
04
-0
.20
91
05
0
.06
57
36
21
Tab
le II
. Con
tinue
d
0.4
80
00
0
0.4
79
85
7
0.0
78
25
6
2.8
79
13
9
0.4
69
53
7
-0.0
63
24
2
0
.48
01
43
-0
.03
41
80
2
.88
08
58
-0
.20
50
80
0
.06
93
11
0.4
90
00
0
0.4
89
92
9
0.0
77
59
1
2.9
39
57
4
0.4
65
54
7
-0.0
68
79
6
0
.49
00
71
-0
.03
34
77
2
.94
04
23
-0
.20
08
61
0
.07
23
54
0.5
00
00
0
0.5
00
00
0
0.0
76
86
9
3.0
00
00
0
0.4
61
21
5
0.0
00
00
0
0
.50
00
00
-0
.03
27
42
3
.00
00
00
-0
.19
64
53
0
.00
00
00
0.5
10
00
0
0.5
10
06
9
0.0
76
08
9
3.0
60
41
3
0.4
56
53
5
-0.0
79
96
9
0
.50
99
31
-0
.03
19
75
3
.05
95
87
-0
.19
18
48
0
.07
84
96
0.5
20
00
0
0.5
20
13
6
0.0
75
26
0
3.1
20
81
3
0.4
51
55
8
-0.0
84
98
2
0
.51
98
64
-0
.03
11
83
3
.11
91
86
-0
.18
71
01
0
.08
09
77
0.5
30
00
0
0.5
30
20
0
0.0
74
37
8
3.1
81
20
1
0.4
46
26
8
-0.0
90
24
6
0
.52
98
00
-0
.03
03
66
3
.17
87
98
-0
.18
21
93
0
.08
37
03
0.5
40
00
0
0.5
40
26
3
0.0
73
44
5
3.2
41
57
6
0.4
40
67
2
-0.0
95
03
7
0
.53
97
37
-0
.02
95
22
3
.23
84
23
-0
.17
71
32
0
.08
59
40
0.5
50
00
0
0.5
50
32
3
0.0
72
46
4
3.3
01
93
7
0.4
34
78
6
-0.0
99
88
6
0
.54
96
77
-0
.02
86
56
3
.29
80
62
-0
.17
19
37
0
.08
82
21
0.5
60
00
0
0.5
60
38
1
0.0
71
43
9
3.3
62
28
5
0.4
28
63
4
-0.1
03
93
1
0
.55
96
19
-0
.02
77
71
3
.35
77
15
-0
.16
66
28
0
.08
96
74
0.5
70
00
0
0.5
70
43
6
0.0
70
37
0
3.4
22
61
7
0.4
22
21
8
-0.1
08
78
9
0
.56
95
64
-0
.02
68
69
3
.41
73
82
-0
.16
12
11
0
.09
19
33
0.5
80
00
0
0.5
80
48
9
0.0
69
25
3
3.4
82
93
4
0.4
15
51
6
-0.1
13
23
8
0
.57
95
11
-0
.02
59
44
3
.47
70
65
-0
.15
56
63
0
.09
37
57
0.5
90
00
0
0.5
90
53
9
0.0
68
09
5
3.5
43
23
5
0.4
08
56
9
-0.1
17
19
3
0
.58
94
61
-0
.02
50
05
3
.53
67
64
-0
.15
00
28
0
.09
50
62
0.6
00
00
0
0.6
00
58
7
0.0
66
89
5
3.6
03
52
1
0.4
01
37
2
-0.1
21
49
0
0
.59
94
13
-0
.02
40
50
3
.59
64
78
-0
.14
43
00
0
.09
66
86
0.6
10
00
0
0.6
10
63
2
0.0
65
65
7
3.6
63
79
0
0.3
93
94
0
-0.1
25
09
1
0
.60
93
68
-0
.02
30
83
3
.65
62
09
-0
.13
84
97
0
.09
75
83
0.6
20
00
0
0.6
20
67
4
0.0
64
38
1
3.7
24
04
2
0.3
86
28
7
-0.1
28
98
7
0
.61
93
26
-0
.02
21
05
3
.71
59
56
-0
.13
26
33
0
.09
87
47
0.6
30
00
0
0.6
30
71
3
0.0
63
06
7
3.7
84
27
7
0.3
78
40
1
-0.1
32
89
8
0
.62
92
87
-0
.02
11
16
3
.77
57
21
-0
.12
66
98
0
.09
98
95
0.6
40
00
0
0.6
40
74
9
0.0
61
71
5
3.8
44
49
5
0.3
70
29
2
-0.1
36
06
4
0
.63
92
51
-0
.02
01
17
3
.83
55
03
-0
.12
07
05
0
.10
02
59
0.6
50
00
0
0.6
50
78
3
0.0
60
33
5
3.9
04
69
6
0.3
62
00
7
-0.1
39
57
6
0
.64
92
17
-0
.01
91
17
3
.89
53
02
-0
.11
47
01
0
.10
09
33
0.6
60
00
0
0.6
60
81
3
0.0
58
91
6
3.9
64
87
8
0.3
53
49
8
-0.1
43
04
5
0
.65
91
87
-0
.01
81
07
3
.95
51
20
-0
.10
86
41
0
.10
15
23
0.6
70
00
0
0.6
70
84
0
0.0
57
46
4
4.0
25
04
1
0.3
44
78
6
-0.1
46
43
4
0
.66
91
60
-0
.01
70
91
4
.01
49
57
-0
.10
25
48
0
.10
19
87
0.6
80
00
0
0.6
80
86
4
0.0
55
98
3
4.0
85
18
6
0.3
35
90
0
-0.1
49
20
4
0
.67
91
35
-0
.01
60
76
4
.07
48
12
-0
.09
64
54
0
.10
17
84
0.6
90
00
0
0.6
90
88
5
0.0
54
47
1
4.1
45
31
1
0.3
26
82
6
-0.1
52
36
6
0
.68
91
14
-0
.01
50
58
4
.13
46
86
-0
.09
03
47
0
.10
19
21
0.7
00
00
0
0.7
00
90
3
0.0
52
93
1
4.2
05
41
8
0.3
17
58
5
-0.1
55
37
9
0
.69
90
97
-0
.01
40
42
4
.19
45
80
-0
.08
42
52
0
.10
18
51
0.7
10
00
0
0.7
10
91
7
0.0
51
36
0
4.2
65
50
4
0.3
08
16
0
-0.1
58
10
1
0
.70
90
82
-0
.01
30
26
4
.25
44
94
-0
.07
81
55
0
.10
14
31
0.7
20
00
0
0.7
20
92
8
0.0
49
76
5
4.3
25
57
1
0.2
98
59
3
-0.1
60
66
1
0
.71
90
71
-0
.01
20
17
4
.31
44
26
-0
.07
21
01
0
.10
07
83
0.7
30
00
0
0.7
30
93
6
0.0
48
14
3
4.3
85
61
7
0.2
88
86
0
-0.1
63
52
3
0
.72
90
63
-0
.01
10
12
4
.37
43
80
-0
.06
60
71
0
.10
03
62
0.7
40
00
0
0.7
40
94
1
0.0
46
49
8
4.4
45
64
3
0.2
78
98
6
-0.1
65
52
2
0
.73
90
59
-0
.01
00
16
4
.43
43
54
-0
.06
00
94
0
.09
90
05
0.7
50
00
0
0.7
50
94
1
0.0
44
82
9
4.5
05
64
9
0.2
68
97
5
-0.1
68
07
9
0
.74
90
58
-0
.00
90
30
4
.49
43
49
-0
.05
41
79
0
.09
81
18
0.7
60
00
0
0.7
60
93
9
0.0
43
14
1
4.5
65
63
4
0.2
58
84
6
-0.1
69
83
2
0
.75
90
61
-0
.00
80
58
4
.55
43
63
-0
.04
83
49
0
.09
63
40
0.7
70
00
0
0.7
70
93
3
0.0
41
43
2
4.6
25
59
8
0.2
48
59
5
-0.1
72
03
1
0
.76
90
66
-0
.00
71
01
4
.61
43
98
-0
.04
26
06
0
.09
49
02
0.7
80
00
0
0.7
80
92
4
0.0
39
70
5
4.6
85
54
2
0.2
38
23
3
-0.1
73
65
4
0
.77
90
76
-0
.00
61
61
4
.67
44
55
-0
.03
69
68
0
.09
27
84
0.7
90
00
0
0.7
90
91
1
0.0
37
96
2
4.7
45
46
5
0.2
27
77
3
-0.1
75
51
1
0
.78
90
89
-0
.00
52
43
4
.73
45
32
-0
.03
14
56
0
.09
07
73
0.8
00
00
0
0.8
00
89
4
0.0
36
20
3
4.8
05
36
7
0.2
17
21
8
-0.1
76
88
4
0
.79
91
05
-0
.00
43
46
4
.79
46
30
-0
.02
60
78
0
.08
81
50
0.8
10
00
0
0.8
10
87
5
0.0
34
43
1
4.8
65
24
7
0.2
06
58
6
-0.1
78
06
7
0
.80
91
25
-0
.00
34
77
4
.85
47
49
-0
.02
08
63
0
.08
51
90
0.8
20
00
0
0.8
20
85
1
0.0
32
64
9
4.9
25
10
8
0.1
95
89
6
-0.1
79
23
9
0
.81
91
48
-0
.00
26
39
4
.91
48
88
-0
.01
58
36
0
.08
20
53
0.8
30
00
0
0.8
30
82
5
0.0
30
85
7
4.9
84
94
8
0.1
85
14
1
-0.1
80
05
4
0
.82
91
75
-0
.00
18
34
4
.97
50
48
-0
.01
10
05
0
.07
83
79
0.8
40
00
0
0.8
40
79
5
0.0
29
05
8
5.0
44
76
8
0.1
74
34
9
-0.1
80
85
8
0
.83
92
05
-0
.00
10
68
5
.03
52
28
-0
.00
64
07
0
.07
44
85
0.8
50
00
0
0.8
50
76
1
0.0
27
25
3
5.1
04
56
9
0.1
63
51
8
-0.1
81
33
5
0
.84
92
38
-0
.00
03
42
5
.09
54
27
-0
.00
20
55
0
.07
00
31
0.8
60
00
0
0.8
60
72
5
0.0
25
44
3
5.1
64
35
0
0.1
52
66
0
-0.1
81
83
5
0
.85
92
74
0
.00
03
38
5
.15
56
46
0
.00
20
25
0
.06
53
29
0.8
70
00
0
0.8
70
68
5
0.0
23
63
2
5.2
24
11
3
0.1
41
79
3
-0.1
81
94
3
0
.86
93
14
0
.00
09
66
5
.21
58
83
0
.00
57
97
0
.05
99
30
0.8
80
00
0
0.8
80
64
3
0.0
21
82
0
5.2
83
85
6
0.1
30
91
7
-0.1
82
23
9
0
.87
93
57
0
.00
15
40
5
.27
61
39
0
.00
92
38
0
.05
43
49
0.8
90
00
0
0.8
90
59
7
0.0
20
00
7
5.3
43
58
2
0.1
20
03
9
-0.1
82
24
4
0
.88
94
02
0
.00
20
53
5
.33
64
13
0
.01
23
20
0
.04
80
55
0.9
00
00
0
0.9
00
54
8
0.0
18
19
3
5.4
03
29
0
0.1
09
15
7
-0.1
82
04
2
0
.89
94
51
0
.00
25
03
5
.39
67
06
0
.01
50
15
0
.04
10
47
0.9
10
00
0
0.9
10
49
7
0.0
16
38
4
5.4
62
98
2
0.0
98
30
2
-0.1
81
65
6
0
.90
95
02
0
.00
28
77
5
.45
70
14
0
.01
72
60
0
.03
32
35
0.9
20
00
0
0.9
20
44
3
0.0
14
57
7
5.5
22
65
8
0.0
87
46
2
-0.1
81
50
1
0
.91
95
56
0
.00
31
70
5
.51
73
37
0
.01
90
21
0
.02
48
77
0.9
30
00
0
0.9
30
38
7
0.0
12
77
7
5.5
82
32
1
0.0
76
66
5
-0.1
81
00
1
0
.92
96
12
0
.00
33
70
5
.57
76
74
0
.02
02
18
0
.01
52
02
22
Tab
le II
. Con
clud
ed
0.9
40
00
0
0.9
40
32
9
0.0
10
98
1
5.6
41
97
2
0.0
65
88
8
-0.1
80
53
7
0
.93
96
71
0
.00
34
68
5
.63
80
23
0
.02
08
08
0
.00
42
74
0.9
50
00
0
0.9
50
26
9
0.0
09
19
0
5.7
01
61
2
0.0
55
14
2
-0.1
79
67
6
0
.94
97
30
0
.00
34
48
5
.69
83
83
0
.02
06
86
-0
.00
88
02
0.9
60
00
0
0.9
60
20
8
0.0
07
40
8
5.7
61
24
9
0.0
44
44
9
-0.1
78
64
6
0
.95
97
91
0
.00
32
84
5
.75
87
46
0
.01
97
01
-0
.02
46
02
0.9
70
00
0
0.9
70
14
8
0.0
05
63
2
5.8
20
88
9
0.0
33
79
1
-0.1
78
38
1
0
.96
98
51
0
.00
29
46
5
.81
91
06
0
.01
76
77
-0
.04
37
09
0.9
80
00
0
0.9
80
09
0
0.0
03
85
4
5.8
80
54
3
0.0
23
12
6
-0.1
79
62
8
0
.97
99
09
0
.00
23
87
5
.87
94
52
0
.01
43
23
-0
.06
87
51
0.9
90
00
0
0.9
90
03
8
0.0
02
04
3
5.9
40
22
6
0.0
12
26
1
-0.1
86
20
9
0
.98
99
61
0
.00
15
22
5
.93
97
68
0
.00
91
31
-0
.10
67
09
1.0
00
00
0
1.0
00
00
0
0.0
00
00
0
5.9
99
99
7
0.0
00
00
0**
****
****
*
1
.00
00
00
0
.00
00
00
5
.99
99
97
0
.00
00
00
****
****
***
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December 1996 Technical Memorandum
Computer Program To Obtain Ordinates for NACA AirfoilsWU 505-59-10-31
Charles L. Ladson, Cuyler W. Brooks, Jr., Acquilla S. Hill, and Darrell W.Sproles
L-17509
NASA TM-4741
Ladson, Brooks, and Hill: Langley Research Center, Hampton, VA; Sproles: Computer Sciences Corp., Hampton,VA.Computer program is available electronically from the Langley Software Server
Computer programs to produce the ordinates for airfoils of any thickness, thickness distribution, or camber in theNACA airfoil series were developed in the early 1970’s and are published as NASA TM X-3069 and TM X-3284.For analytic airfoils, the ordinates are exact. For the 6-series and all but the leading edge of the 6A-series airfoils,agreement between the ordinates obtained from the program and previously published ordinates is generally within5 × 10−5 chord. Since the publication of these programs, the use of personal computers and individual workstationshas proliferated. This report describes a computer program that combines the capabilities of the previously pub-lished versions. This program is written in ANSI FORTRAN 77 and can be compiled to run on DOS, UNIX, andVMS based personal computers and workstations as well as mainframes. An effort was made to make all inputs tothe program as simple as possible to use and to lead the user through the process by means of a menu.
NACA airfoils; 4-digit; 6-series; Platform-independent; FORTRAN 23
A03
NASA Langley Research CenterHampton, VA 23681-0001
National Aeronautics and Space AdministrationWashington, DC 20546-0001
Unclassified–UnlimitedSubject Category 02Availability: NASA CASI (301) 621-0390
Unclassified Unclassified Unclassified